Wait. Stop.... your plan is upside down a little bit.
Get the content and database moved before you start playing with the template and making a new one. Without the content, you design the template, then copy everything over, then realize that something in the content wasn't accounted for (shortcodes that affect layout like adding columns or in-content tags that will put a box around something or any number of things). So now that you have the content there, you need to go back and change a lot of your CSS and layout work to make all that fit.
Getting a mirror of the site to start with saves you at least a 2-3 hours of dev time because you tackle it as you go rather than tackling it, throwing in a variable (your pages and posts and whatever else), and needing to go back again.
Ideally, you have it planned as a short turnaround. If you are using my "start with something solid and adjust it as needed" technique I talked about last week, it should be a week or so turnaround (so long as the client is motivated). If that's the case, avoid updates as much as possible to the current site. If you must do updates, try to do them on both sites for that week. That will save you headaches and potential pitfalls, too.
Trust me, though - get the content there first. The sooner it's in, the better. (Even of a fresh start, I get the site generally set up so it works and then get the folks doing the content working on it so I've got real content. Once they've had at it for a day or two and have at least one or two pages of each type in there, I can start to work on the design build-out.
As for moving the DB, there are several ways to go about it. A lot depends upon the host. A lot of managed hosts have a tool that will help move the whole site. Any proper backup tool (UpdraftPlus is my go-to) will dump out a proper sql file that you can import into phpMyAdmin. There are also some plugins out there to do the trick. I've heard good things about this one, but haven't used it myself yet since most of my clients this year are moving to managed hosts that have a utility already.
Anyway - change your plan from:
When finished, move DB from old site to new site.
And change it to:
Move DB (and Wp-includes folder) from old site to new site.
Unless you are planning to rework site structure... then it's a little bit different again, though I still like to get the old site there first so I can make sure I've caught all my redirects and pass-throughs.